Question

If the number of uranium nuclei required per hour to produce a power of 64 kW is \(7.2 \times 10^{18}\), then the energy released per fission is

• A. \(0.64 \times 10^{-10}\) J
• B. \(3.2 \times 10^{-13}\) J
• C. \(0.32 \times 10^{-10}\) J
• D. \(3.2 \times 10^{-10}\) J

Hint:

Always ensure your units are consistent before performing calculations. Convert power to Watts (J/s) and time to seconds to get the energy in Joules. This avoids common errors in magnitude.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Power is the rate at which energy is produced. The total energy produced in a given time is the product of the power and the time duration. This total energy is also equal to the number of fissions multiplied by the energy released per fission.
Step 2: Key Formula or Approach:
1. Total Energy, \(E_{total} = \text{Power} (P) \times \text{Time} (t)\).
2. Total Energy, \(E_{total} = \text{Number of fissions} (N) \times \text{Energy per fission} (E_{fission})\).
Equating these gives: \(P \times t = N \times E_{fission}\).
Step 3: Detailed Explanation:
First, convert all given quantities to SI units.
- Power, \(P = 64 \text{ kW} = 64 \times 10^3 \text{ W} = 64 \times 10^3 \text{ J/s}\).
- Time, \(t = 1 \text{ hour} = 60 \times 60 \text{ s} = 3600 \text{ s}\).
- Number of fissions (nuclei) in this time, \(N = 7.2 \times 10^{18}\).
Now, calculate the total energy produced in one hour:
\[ E_{total} = P \times t = (64 \times 10^3 \text{ J/s}) \times (3600 \text{ s}) \] \[ E_{total} = 230400 \times 10^3 \text{ J} = 2.304 \times 10^8 \text{ J} \] This total energy is released from \(N = 7.2 \times 10^{18}\) fissions. We can now find the energy released per fission:
\[ E_{fission} = \frac{E_{total}}{N} = \frac{2.304 \times 10^8 \text{ J}}{7.2 \times 10^{18}} \] \[ E_{fission} = \left(\frac{2.304}{7.2}\right) \times 10^{8-18} \text{ J} \] \[ E_{fission} = 0.32 \times 10^{-10} \text{ J} \] Step 4: Final Answer:
The energy released per fission is \(0.32 \times 10^{-10}\) J. Therefore, option (C) is correct.